It's as Wolf said, whether a parrot bites or not is up to the people who live with the parrot because all species of parrot bite but none is an inherent biter. The difference between a biting and a non-biting parrots is husbandry. Period. Let us know if you need us to elaborate on this and we will.
Now, small species don't bite as hard as the larger ones but, then, smaller species are not usually as cuddly as some (not even all) the larger ones. Ohhh, people tell you their plet, lovie, tiel, etc is, most definitely, a cuddler but you will find that most of these birds are young. Once they become sexually mature and years go by, they become more impatient and aloof and you will see lots and lots of postings asking why is their loving little bird now biting, screaming, flying away from them or plucking. And the answer is that these are all aviary and not companion species. Now, I am sure there are exceptions but one cannot choose a species based on the few and far in between exceptions one might find, right?
Now, for the species that are companion (deeply imprinted to humans) and are cuddly... First thing you need to understand is that cuddly = needy and that means A LOT of hours with the bird on you. In my personal experience, the cuddliest are the cockatoos and the GCCs. Now, the cockatoos are fabulous birds BUT they are big birds and that means a big-ass bird hanging of your neck for hours and hours every single day -not something that most people can put up with for 30 or 40 years. It also means that they have very big and powerful beaks and this means that even when they don't bite you, they can hurt you. My Freddie (Lesser Sulfur) would never, ever, ever bite me no matter what I do to or with him but he uses his beak to hold on to things and, if this 'thing' happens to be your arm, you end up with a good black and blue on it. Same with his claws, if he feels insecure and grabs your hand real hard, you end up with little holes in it. The other problem is that when they are mate-bonded to you, they will 'warn' you of what they perceive as danger or competition to their relationship with you by nipping you. They don't mean it as aggression, it's a natural behavior they use in the wild but they do it to another bird which is covered by thick plumage so the hard nip doesn't hurt at all while, we, humans, are terribly inadequate and naked birds and can get hurt too easily. They are also TERRIBLY destructive and LOUD LOUD LOUD! A couple of months ago he made a flock call while on my shoulder and I could actually feel my brain vibrating inside my skull - that's how powerful their vocalizations are. He was given up because he was a screamer (his owners worked outside the house during the day so he screamed and screamed all day long) and it took me ten months to break him out of it -and that's another problem with toos that are not paid enough attention, as well as plucking and even self-mutilating. There is a good reason why toos are the number one given up bird...
Now, people will tell you that GCCs are moody and nippy but, in my personal experience, this is simply not true. I've taken in four of them, all given up because of biting, but out of the four, only one was really bitey and it was a male that had been abused and severely neglected. The other three became sweet-tempered in a matter of a few weeks because the only problem they had was inadequate husbandry. I rehomed three of the four because this is a species that takes up A LOT of one-on-one and I simply did not have the time for them (I care for a lot of animals and I am always spreading myself thin as it is). The fourth one I still have. Her name is Codee and she is the sweetest, sweetest thing! She never screams, comes running when I call her so she can jump on my arm, quickly climbs up to my shoulder and kisses me on the cheek (she also kisses me -puts her beak on my cheek and goes SMACK SMACK SMACK

- when I kiss her head or tell her I love her) and never, ever, ever bites. So much so that she is one of the few birds that I trust with my grandkids! BUT this is a bird that lives cage-free, is kept at a strict solar schedule, is fed a good fresh food diet with low protein, has a mate of her own for 24/7/365 company and spends a solid two hours on me every single day! And that is what I meant by husbandry been the key to a non-biting parrot.
I don't know if I answered your question to your satisfaction but, please, let me know if you have any questions or doubts or need any clarification.